An inequality for the pedal simplex (Q5946236)

Let \(S\) denote a \(d\)-dimensional simplex in Euclidean \(d\)-space, and \(\overline S\) be the pedal simplex of \(S\) with respect to some interior point \(x\) of \(S\) (i.e., \(\overline S\) is the convex hull of the \(d+1\) perpendicular footpoints of \(x\) within the facet hyperplanes of \(S\)). Inspired by the known inequality \[ V(\overline{S})\leq \frac{1}{d^d} V(S)\tag{*} \] for the volumes of \(\overline{S}\) and \(S\), the author gives a sequence of inequalities (containing also informations on two-dimensional simplices spanned by \(x\) and, in each case, two perpendicular footpoints) which yield refinements of (*). The cases of equality are discussed in terms of dihedral angles of facets and of barycentric coordinates of \(x\).